zlaqr2()

subroutine zlaqr2	(	logical	wantt,
		logical	wantz,
		integer	n,
		integer	ktop,
		integer	kbot,
		integer	nw,
		complex*16, dimension( ldh, * )	h,
		integer	ldh,
		integer	iloz,
		integer	ihiz,
		complex*16, dimension( ldz, * )	z,
		integer	ldz,
		integer	ns,
		integer	nd,
		complex*16, dimension( * )	sh,
		complex*16, dimension( ldv, * )	v,
		integer	ldv,
		integer	nh,
		complex*16, dimension( ldt, * )	t,
		integer	ldt,
		integer	nv,
		complex*16, dimension( ldwv, * )	wv,
		integer	ldwv,
		complex*16, dimension( * )	work,
		integer	lwork )

ZLAQR2 performs the unitary similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation).

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Purpose:

!>
!>    ZLAQR2 is identical to ZLAQR3 except that it avoids
!>    recursion by calling ZLAHQR instead of ZLAQR4.
!>
!>    Aggressive early deflation:
!>
!>    ZLAQR2 accepts as input an upper Hessenberg matrix
!>    H and performs an unitary similarity transformation
!>    designed to detect and deflate fully converged eigenvalues from
!>    a trailing principal submatrix.  On output H has been over-
!>    written by a new Hessenberg matrix that is a perturbation of
!>    an unitary similarity transformation of H.  It is to be
!>    hoped that the final version of H has many zero subdiagonal
!>    entries.
!>
!> 

Parameters

[in]	WANTT	!> WANTT is LOGICAL !> If .TRUE., then the Hessenberg matrix H is fully updated !> so that the triangular Schur factor may be !> computed (in cooperation with the calling subroutine). !> If .FALSE., then only enough of H is updated to preserve !> the eigenvalues. !>
[in]	WANTZ	!> WANTZ is LOGICAL !> If .TRUE., then the unitary matrix Z is updated so !> so that the unitary Schur factor may be computed !> (in cooperation with the calling subroutine). !> If .FALSE., then Z is not referenced. !>
[in]	N	!> N is INTEGER !> The order of the matrix H and (if WANTZ is .TRUE.) the !> order of the unitary matrix Z. !>
[in]	KTOP	!> KTOP is INTEGER !> It is assumed that either KTOP = 1 or H(KTOP,KTOP-1)=0. !> KBOT and KTOP together determine an isolated block !> along the diagonal of the Hessenberg matrix. !>
[in]	KBOT	!> KBOT is INTEGER !> It is assumed without a check that either !> KBOT = N or H(KBOT+1,KBOT)=0. KBOT and KTOP together !> determine an isolated block along the diagonal of the !> Hessenberg matrix. !>
[in]	NW	!> NW is INTEGER !> Deflation window size. 1 <= NW <= (KBOT-KTOP+1). !>
[in,out]	H	!> H is COMPLEX*16 array, dimension (LDH,N) !> On input the initial N-by-N section of H stores the !> Hessenberg matrix undergoing aggressive early deflation. !> On output H has been transformed by a unitary !> similarity transformation, perturbed, and the returned !> to Hessenberg form that (it is to be hoped) has some !> zero subdiagonal entries. !>
[in]	LDH	!> LDH is INTEGER !> Leading dimension of H just as declared in the calling !> subroutine. N <= LDH !>
[in]	ILOZ	!> ILOZ is INTEGER !>
[in]	IHIZ	!> IHIZ is INTEGER !> Specify the rows of Z to which transformations must be !> applied if WANTZ is .TRUE.. 1 <= ILOZ <= IHIZ <= N. !>
[in,out]	Z	!> Z is COMPLEX*16 array, dimension (LDZ,N) !> IF WANTZ is .TRUE., then on output, the unitary !> similarity transformation mentioned above has been !> accumulated into Z(ILOZ:IHIZ,ILOZ:IHIZ) from the right. !> If WANTZ is .FALSE., then Z is unreferenced. !>
[in]	LDZ	!> LDZ is INTEGER !> The leading dimension of Z just as declared in the !> calling subroutine. 1 <= LDZ. !>
[out]	NS	!> NS is INTEGER !> The number of unconverged (ie approximate) eigenvalues !> returned in SR and SI that may be used as shifts by the !> calling subroutine. !>
[out]	ND	!> ND is INTEGER !> The number of converged eigenvalues uncovered by this !> subroutine. !>
[out]	SH	!> SH is COMPLEX*16 array, dimension (KBOT) !> On output, approximate eigenvalues that may !> be used for shifts are stored in SH(KBOT-ND-NS+1) !> through SR(KBOT-ND). Converged eigenvalues are !> stored in SH(KBOT-ND+1) through SH(KBOT). !>
[out]	V	!> V is COMPLEX*16 array, dimension (LDV,NW) !> An NW-by-NW work array. !>
[in]	LDV	!> LDV is INTEGER !> The leading dimension of V just as declared in the !> calling subroutine. NW <= LDV !>
[in]	NH	!> NH is INTEGER !> The number of columns of T. NH >= NW. !>
[out]	T	!> T is COMPLEX*16 array, dimension (LDT,NW) !>
[in]	LDT	!> LDT is INTEGER !> The leading dimension of T just as declared in the !> calling subroutine. NW <= LDT !>
[in]	NV	!> NV is INTEGER !> The number of rows of work array WV available for !> workspace. NV >= NW. !>
[out]	WV	!> WV is COMPLEX*16 array, dimension (LDWV,NW) !>
[in]	LDWV	!> LDWV is INTEGER !> The leading dimension of W just as declared in the !> calling subroutine. NW <= LDV !>
[out]	WORK	!> WORK is COMPLEX*16 array, dimension (LWORK) !> On exit, WORK(1) is set to an estimate of the optimal value !> of LWORK for the given values of N, NW, KTOP and KBOT. !>
[in]	LWORK	!> LWORK is INTEGER !> The dimension of the work array WORK. LWORK = 2*NW !> suffices, but greater efficiency may result from larger !> values of LWORK. !> !> If LWORK = -1, then a workspace query is assumed; ZLAQR2 !> only estimates the optimal workspace size for the given !> values of N, NW, KTOP and KBOT. The estimate is returned !> in WORK(1). No error message related to LWORK is issued !> by XERBLA. Neither H nor Z are accessed. !>

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA